Let (N, g) be a Riemannian manifold with the distance function d(x, y) and an open subset M ⊂ N . For x ∈ M we denote by D x the distance difference function DWe consider the inverse problem of determining the topological and the differentiable structure of the manifold M and the metric g| M on it when we are given the distance difference data, that is, the set F , the metric g| F , and the collection D(M ) = {D x ; x ∈ M }. Moreover, we consider the embedded image D(M ) of the manifold M , in the vector space C(F × F ), as a representation of manifold M . The inverse problem of determining (M, g) from D(M ) arises e.g. in the study of the wave equation on R×N when we observe in F the waves produced by spontaneous point sources at unknown points (t, x) ∈ R × M . Then D x (z 1 , z 2 ) is the difference of the times when one observes at points z 1 and z 2 the wave produced by a point source at x that goes off at an unknown time. The problem has applications in hybrid inverse problems and in geophysical imaging.

Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation ∂ 2 t u − ∆ g u = χW , where W is a random variable with the white noise statistics on R 1+n , n ≥ 3, χ is a smooth function vanishing for negative times and outside a compact set in space, and ∆ g is the Laplace-Beltrami operator associated to a smooth non-trapping Riemannian metric tensor g on R n . The metric tensor g models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set X ⊂ R n ,for T > 0. Supposing that χ is non-zero on X and constant in time after t > 1, we show that in the limit T → ∞, the data C T becomes statistically stable, that is, independent of the realization of W . Our main result is that, with probability one, this limit determines the Riemannian manifold (R n , g) up to an isometry. To our knowledge, this is the first result showing that a medium can be determined in a passive imaging setting, without assuming a separation of scales.Date: May 7, 2019.

E. Iversen et al. SUMMARYWith a Hamilton-Jacobi equation in Cartesian coordinates as a starting point, it is common to use a system of ordinary differential equations describing the continuation of first-order phase-space perturbation derivatives along a reference ray. Such derivatives can be exploited for calculation of geometrical spreading on the reference ray, and for establishing a framework for second-order extrapolation of traveltime to points outside the reference ray. The continuation of the first-order phase-space perturbation derivatives has historically been referred to as dynamic ray tracing.The reason for this is its importance in the process of calculating amplitudes along the reference ray. We extend the standard dynamic ray tracing scheme to include higher orders in the phase-space perturbation derivatives. The main motivation is to extrapolate and interpolate important amplitude and phase properties of highfrequency Green's functions with better accuracy. Principal amplitude coefficients, geometrical spreading factors, traveltimes, slowness vectors, and curvature matrices are examples of quantities for which we enhance the computation potential. This, in turn, has immediate applications in modelling, mapping, and imaging. Numerical tests for 3D isotropic and anisotropic heterogeneous models yield clearly improved extrapolation results for traveltime and geometrical spreading. One important conclusion is that the extrapolation function for geometrical spreading must be at least third order to be appropriate at large distances away from the reference ray. E. Iversen et al.The developed methodology has the following main applications:• Fast computation of high-frequency elastic-wave Green's functions through (Hermite or spline) interpolation and extrapolation of amplitude and phase with derivatives. Our procedure holds in generally anisotropic media, leading to systems of equations describing the propagation of elastic waves, of principal type.